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<p><dfn class="terminology">Comments</dfn>: For the Euler’s equation <span class="process-math">\(x^2 y^{\prime \prime}+\alpha x y^{\prime}+\beta y=0\text{,}\)</span>(i) <span class="process-math">\(x=0\)</span> is a regular singular point;(ii) According to (<a href="" class="xref" data-knowl="./knowl/eq5_8.html" title="Equation 5.4.1">(5.4.1)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq5_9.html" title="Equation 5.4.3">(5.4.3)</a>), one solution is <span class="process-math">\(y_1=x^{r_1}\)</span> (<span class="process-math">\(r_1\)</span> is not necessarily positive integers).(iii) This suggests that for a regular singular point, we may seek a solution of the form:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_8.html ./knowl/eq5_9.html">
\begin{equation*}
y=x^r \sum_{n=0}^{\infty} a_n x^n=\sum_{n=0}^{\infty} a_n x^{n+r},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(r\)</span> is to be determined. More generally, one may set</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_8.html ./knowl/eq5_9.html">
\begin{equation*}
y=\sum_{n=0}^{\infty} a_n (x-x_0)^{n+r}.
\end{equation*}
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